This is very slow
Condition34=Reduce[{(-a+c β+t θ1)^2(6b^2+4b λ+λ^2)/(8b^3)>
(3a^2-6a c α+3c^2 α^2+2a(-3+2p)t θ2+2c(3-2p)t α θ2+(3-4p+2p^2)t^2 θ2^2)/(4b),
a>c>0,1>p>0,1>t>0,1>b>0,1>λ>0,β>α>1,a>c α,a>c β,0<θ1<θ2<θ3<1},β]
Verify we can divide both sides
Simplify[(6b^2+4b λ+λ^2)/(8b^3)>0,
{a>c>0,1>p>0,1>t>0,1>b>0,1>λ>0,β>α>1,a>c α,a>c β,0<θ1<θ2<θ3<1}]
returns
True
Try to make the problem easier, but this is still very slow
Condition34=Reduce[{(-a+c β+t θ1)^2>
(3a^2-6a c α+3c^2 α^2+2a(-3+2p)t θ2+2c(3-2p)t α θ2+(3-4p+2p^2)t^2 θ2^2)/(4b)/(6b^2+4b λ+λ^2)*(8b^3),
a>c>0,1>p>0,1>t>0,1>b>0,1>λ>0,β>α>1,a>c α,a>c β,0<θ1<θ2<θ3<1},β]
Try to make the problem easier
Condition34=Reduce[{(-a+c β+t θ1)^2>q,
a>c>0,1>p>0,1>t>0,1>b>0,1>λ>0,β>α>1,a>c α,a>c β,0<θ1<θ2<θ3<1},β]/.
q->(3a^2-6a c α+3c^2 α^2+2a(-3+2p)t θ2+2c(3-2p)t α θ2+(3-4p+2p^2)t^2 θ2^2)/(4b)/(6b^2+4b λ+λ^2)*(8b^3)
The output is very large because we have not applied any domain information
This is still slow, but it may provide a more compact result for you
Simplify[Condition34,{a>c>0,1>p>0,1>t>0,1>b>0,1>λ>0,β>α>1,a>c α,a>c β,0<θ1<θ2<θ3<1}]
This
(Reduce[{(-a+c β+t θ1)^2>q,
a>c>0,1>p>0,1>t>0,1>b>0,1>λ>0,β>α>1,a>c α,a>c β,0<θ1<θ2<θ3<1},β]/.
q->(3a^2-6a c α+3c^2 α^2+2a(-3+2p)t θ2+2c(3-2p)t α θ2+(3-4p+2p^2)t^2 θ2^2)/(4b)/(6b^2+4b λ+λ^2)*(8b^3))/.
{Inequality[0, Less, c, Less, a]->True,Inequality[0, Less, p, Less, 1]->True,
Inequality[0, Less, t, Less, 1]->True,Inequality[0, Less, b, Less, 1]->True,
Inequality[0, Less, λ, Less, 1]->True,Inequality[0, Less, a, LessEqual, 1]->True,
Inequality[θ1, Less, θ2, Less, 1]->True,Inequality[θ2, Less, θ3, Less, 1]->True,
Inequality[α, Less, β, Less, a/c]->True,Inequality[0, Less, θ1, Less, 1]->True,
Inequality[1, Less, α, Less, a/c]->True
}
will be much faster than Simplify, but it will not make the result as simple and it is necessary to be very careful with the pattern matching to make certain the replacements are exactly correct. If you look at the result of this there will still be many inequalities that might be replaced with True and thus eliminated if it is possible to justify the replacements. The result is still very large, but it might be possible to use this method to get it to be small enough that Simplify could then be done in acceptable time.
Please check all this very carefully.